Problem 62 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(y+0.52<1.05 y\) or \(9.8-15 y>-15.7\)

Step-by-Step Solution

Answer

\((- finity, 1.7) \cup (10.4, finity)\).

Step 1: Simplify the First Inequality

Start with the inequality \(y + 0.52 < 1.05y\). Subtract \(y\) from both sides to get \(0.52 < 0.05y\). Finally, divide both sides by 0.05 to solve for \(y\), yielding \(y > 10.4\).

Step 2: Simplify the Second Inequality

Take the inequality \(9.8 - 15y > -15.7\). First, subtract 9.8 from both sides to get \(-15y > -25.5\). Then, divide both sides by -15 (this flips the inequality sign) to get \(y < 1.7\).

Step 3: Find the Union of the Solutions

It's important to remember that we are looking for solutions to either of the inequalities since the compound inequality is connected by an "or." The solutions are \(y > 10.4\) or \(y < 1.7\). The union includes all \(y\) values outside the interval [1.7, 10.4].

Step 4: Graph the Solution Set

On a number line, mark \(1.7\) with an open circle and shade to the left, to represent \(y < 1.7\). Also, mark \(10.4\) with an open circle, and shade to the right, to indicate \(y > 10.4\).

Step 5: Write Solution in Interval Notation

Since the solution set consists of all \(y\) values less than 1.7 or greater than 10.4, the interval notation is \((-finity, 1.7) \cup (10.4, finity)\).

Key Concepts

Interval Notation

Interval notation is a shorthand way to describe a set of numbers along the real number line. It’s a compact way of expressing sets, especially useful when dealing with inequalities.
When you are working with the solution to a compound inequality, like in the original exercise, interval notation can succinctly communicate which values are included in the solution set.

• The symbol \( ( \) denotes that the endpoint is not included in the solution set, known as an open interval.
• Conversely, \( [ \) denotes that the endpoint is included, known as a closed interval.
• Union, represented as \( \cup \), is used to combine intervals.

In our exercise, we have the solutions \( y < 1.7 \) or \( y > 10.4 \). This translates to the interval notation \((-\infty, 1.7) \cup (10.4, \infty)\). This means that any number less than 1.7 or greater than 10.4 is part of the solution set. This concise format helps simplify complex expressions and allows easy communication of ranges of values.

Number Line Graphing

Number line graphing is a visual representation of the inequalities. It helps you see which values satisfy the inequality and where they lie on the number line. For the compound inequality from the original exercise, graphing makes it clear where the solutions to the inequalities lie.
To draw this on a number line, follow these steps:

• First, identify the critical points, which in this case are \( 1.7 \) and \( 10.4 \).
• Place open circles on these critical points. Open circles indicate that these numbers themselves are not included in the solution.
• Shade the portion of the line to the left of \( 1.7 \) to show \( y < 1.7 \).
• Shade the portion of the line to the right of \( 10.4 \) to represent \( y > 10.4 \).

The portions of the number line shaded in this graph represent all numbers \( y \) that satisfy the compound inequality. Graphing is effective because it visually displays the solution regions and makes the relationship between numbers more intuitive.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that make the inequality true. In compound inequalities, you have more than one inequality constraint that determines the solution set.
The process of solving inequalities is similar to solving equations, but with some crucial differences, particularly involving multiplication or division by negative numbers:

• When we subtract or add the same number to both sides, the inequality sign does not change.
• Multiplying or dividing both sides by a positive number does not change the inequality sign.
• However, multiplying or dividing both sides by a negative number reverses the inequality sign.

In the original exercise, solving the first inequality \( y + 0.52 < 1.05y \) involved subtracting \( y \) from both sides and dividing by \( 0.05 \). The second inequality \( 9.8 - 15y > -15.7 \) required subtracting \( 9.8 \), then dividing by \(-15\), which reversed the inequality. Understanding these steps and rules helps solve compound inequalities correctly and effectively.